1. Field of the Invention
The present invention relates to performing arithmetic operations on interval operands within a computer system. More specifically, the present invention relates to a method and an apparatus for using a computer system to find the roots of a polynomial equation with interval coefficients.
2. Related Art
Rapid advances in computing technology make it possible to perform trillions of computational operations each second. This tremendous computational speed makes it practical to perform computationally intensive tasks as diverse as predicting the weather and optimizing the design of an aircraft engine. Such computational tasks are typically performed using machine-representable floating-point numbers to approximate values of real numbers. (For example, see the Institute of Electrical and Electronics Engineers (IEEE) standard 754 for binary floating-point numbers.)
In spite of their limitations, floating-point numbers are generally used to perform most computational tasks.
One limitation is that machine-representable floating-point numbers have a fixed-size word length, which limits their accuracy. Note that a floating-point number is typically encoded using a 32, 64 or 128-bit binary number, which means that there are only 232, 264 or 2128 possible symbols that can be used to specify a floating-point number. Hence, most real number values can only be approximated with a corresponding floating-point number. This creates estimation errors that can be magnified through even a few computations, thereby adversely affecting the accuracy of a computation.
A related limitation is that floating-point numbers contain no information about their accuracy. Most measured data values include some amount of error that arises from the measurement process itself. This error can often be quantified as an accuracy parameter, which can subsequently be used to determine the accuracy of a computation. However, floating-point numbers are not designed to keep track of accuracy information, whether from input data measurement errors or machine rounding errors. Hence, it is not possible to determine the accuracy of a computation by merely examining the floating-point number that results from the computation.
Interval arithmetic has been developed to solve the above-described problems. Interval arithmetic represents numbers as intervals specified by a first (left) endpoint and a second (right) endpoint. For example, the interval [a, b], where a<b, is a closed, bounded subset of the real numbers, R, which includes a and b as well as all real numbers between a and b. Arithmetic operations on interval operands (interval arithmetic) are defined so that interval results always contain the entire set of possible values. The result is a mathematical system for rigorously bounding numerical errors from all sources, including measurement data errors, machine rounding errors and their interactions.
Note that the first endpoint normally contains the “infimum”, which is the largest number that is less than or equal to each of a given set of real numbers. Similarly, the second endpoint normally contains the “supremum”, which is the smallest number that is greater than or equal to each of the given set of real numbers. Also note that within the following disclosure, the infimum of an interval A can be represented as either AL or A. Similarly, the supremum can be represented as either AU or Ā.
When solving systems of nonlinear equations and optimization problems, we frequently want to compute the real roots of a polynomial equation in which the coefficients are intervals. These roots may be finite intervals, semi-infinite intervals, or the entire real line. A naïve procedure for determining the roots can be surprisingly complicated. Consider the quadratic equation Ax2+Bx+C=0 where A=[AL, AU], B=[BL, BU] and C=[CL, CU] are intervals. The interval roots of this quadratic equation are the set of real roots x of the quadratic equation ax2+bx+c=0 for all a∈A, b∈B and c∈C.
When the coefficients are degenerate intervals, we can express the roots as:       r    ±    =                              -          B                ±                              (                                          B                2                            -                              4                ⁢                A                ⁢                                                                   ⁢                C                                      )                                1            /            2                                      2        ⁢        A              .  
If we compute interval roots in this way, they are not sharp except in special cases. This is because the intervals A and B occur more than once in this expression and dependence causes loss of sharpness. It does not help to write the roots in the algebraically equivalent form       r    ±    =                    2        ⁢        C                              -          B                ±                              (                                          B                2                            -                              4                ⁢                A                ⁢                                                                   ⁢                C                                      )                                1            /            2                                .  
What is needed is a method and an apparatus for computing the roots quadratic and polynomial equations with interval coefficients but without the problem of losing sharpness due to dependence.